Complutense University Library

On the Picard Group of Low-codimension Subvarieties


Arrondo Esteban, Enrique and Caravantes, Jorge (2009) On the Picard Group of Low-codimension Subvarieties. Indiana University Mathematics Journal, 58 (3). pp. 1023-1042. ISSN 0022-2518

[img] PDF
Restringido a Repository staff only hasta 2020.


Official URL:


We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n - 2 inherits the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (LIP to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.

Item Type:Article
Uncontrolled Keywords:Projective spaces; Manifolds; Low codimension; Picard group
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14769

[1] E. Arrondo, Subcanonicity of codimension two subvarieties, Rev. Mat. Compl. 18 (2005), 69–80.

[2] E, Arrondo, M.L. Fania, Evidence to subcanonicity of codimension two subvarieties of G(1, 4), to appear

on Int. J. Math.

[3] W. Barth, Transplanting cohomology classes in complex projective space, Amer. J. Math. 92 (1970),


[4] W. Barth, M.E. Larsen, On the homotopy-groups of complex projective manifolds, Math Scand. 30 (1972),


[5] W. Barth and A. Van de Ven, On the geometry in codimension 2 of Grassmann manifolds, Lecture Notes

in Math. 412, Springer Verlag (1974), 1–35.

[6] O. Debarre, Th´eor`emes de connexit´e pour les produits d’espaces projectifs et les grassmanniennes, Amer.

J. Math. 118 (1996), no. 6, 1347–1367.

[7] R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974),


[8] S.L. Kleiman, D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082.

[9] M.E. Larsen, On the topology of projective manifolds, Invent. Math. 19 (1973), 251–260.

[10] A.J. Sommese, Complex subspaces of homogeneous complex manifolds. II. Homotopy results, Nagoya

Math. J. 86, 101–129.

Deposited On:18 Apr 2012 08:26
Last Modified:06 Feb 2014 10:08

Repository Staff Only: item control page